Cosine of 45 Degrees

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Theorem

$\cos 45 \degrees = \cos \dfrac \pi 4 = \dfrac {\sqrt 2} 2$

where $\cos$ denotes the cosine.


Proof

\(\ds \paren {\cos 45 \degrees}^2\) \(=\) \(\ds 1 - \paren {\sin 45 \degrees}^2\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds 1 - \paren {\frac {\sqrt 2} 2}^2\) Sine of $45 \degrees$
\(\ds \) \(=\) \(\ds \frac 1 2\)
\(\ds \leadsto \ \ \) \(\ds \cos 45 \degrees\) \(=\) \(\ds \sqrt {\frac 1 2}\) positive because $\cos 45 \degrees$ is in Quadrant I
\(\ds \) \(=\) \(\ds \dfrac {\sqrt 2} 2\)

$\blacksquare$


Also presented as

Some sources present the cosine of $45 \degrees$ as:

$\cos 45 \degrees = \dfrac 1 {\sqrt 2}$


Sources

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